Editing Conway group (section)

{{Group theory sidebar |Finite}}

In the area of modern algebra known as [[group theory]], the '''Conway groups''' are the three [[sporadic simple group]]s [[Conway group Co1|Co<sub>1</sub>]], [[Conway group Co2|Co<sub>2</sub>]] and [[Conway group Co3|Co<sub>3</sub>]] along with the related finite group [[Leech lattice#Symmetries|Co<sub>0</sub>]] introduced by {{harvs |authorlink=John Horton Conway |last=Conway |year1=1968 |year2=1969}}.

The largest of the Conway groups, '''Co<sub>0</sub>''', is the [[Automorphism group|group of automorphisms]] of the [[Leech lattice]] Λ with respect to addition and [[inner product]].  It has [[Order (group theory)|order]]

: {{val|fmt=commas|8,315,553,613,086,720,000}}

but it is not a simple group.  The simple group '''[[Conway group Co1|Co<sub>1</sub>]]''' of order

: {{val|fmt=commas|4,157,776,806,543,360,000}} = &nbsp;2<sup>21</sup>{{·}}3<sup>9</sup>{{·}}5<sup>4</sup>{{·}}7<sup>2</sup>{{·}}11{{·}}13{{·}}23

is defined as the quotient of '''Co<sub>0</sub>''' by its [[center of a group|center]], which consists of the scalar matrices ±1. The groups '''[[Conway group Co2|Co<sub>2</sub>]]''' of order

:{{val|fmt=commas|42,305,421,312,000}} = &nbsp;2<sup>18</sup>{{·}}3<sup>6</sup>{{·}}5<sup>3</sup>{{·}}7{{·}}11{{·}}23

and '''[[Conway group Co3|Co<sub>3</sub>]]''' of order

:{{val|fmt=commas|495,766,656,000}} = &nbsp;2<sup>10</sup>{{·}}3<sup>7</sup>{{·}}5<sup>3</sup>{{·}}7{{·}}11{{·}}23
 
consist of the automorphisms of Λ fixing a lattice vector of type 2 and type 3, respectively.  As the scalar &minus;1 fixes no non-zero vector, these two groups are isomorphic to subgroups of Co<sub>1</sub>.

The '''inner product''' on the Leech lattice is defined as 1/8 the [[dot product|sum of the products]] of respective co-ordinates of the two multiplicand vectors; it is an integer. The '''square norm''' of a vector is its inner product with itself, always an even integer. It is common to speak of the '''type''' of a Leech lattice vector: half the square norm. Subgroups are often named in reference to the ''types'' of relevant fixed points.  This lattice has no vectors of type 1.